Fuel filter first!

The symptoms are similar to a vacuum leak - it results in the air / fuel
ratio being too lean. The cold engine gets extra fuel to start up and,
as you say, that cuts off after a minute or so. Then the motor ought to
run normally but with a vacuum leak yours stumbles, especially at idle,
as it warms up. Once warm, the vacuum leak is eliminated.

I suggest you check the intake manifold gasket where the manifold is
bolted to the cylinder heads. You may hear a hissing noise but more
likely the leak will be found by CAREFULLY applying propane or fuel to
any suspected leak site. That's dangerous so don't put your face near
that fuel!

Fuel injector on your car is simply a nozzle. However, I don't think it is
your fuel injector at all. I think your problem is the temperature sensor
for the fuel injection system is a bit faulty and at this age, I would
change them out.

I don't think it is leaky intake manifold either, but you could have leaky
vacuum system. Easy way to check this out is by spraying carburator cleaner
on all part of engine and hear if your engine changes at the affected area.

The fuel mixture works in open loop mode (controls the mixture without
feedback), until the oxygen sensor is warm after approx. one minute,
then it goes into closed loop mode (controls the mixture with
feedback).

Until engine has reached normal operating temperature, it will
normally provide richer mixture. Apparently this does not happen in
your case, but it works fine when reaching operating temperature.

So, I agree with Tiger: The ECU gets wrong information about engine
temperature.